Fourth dimension

# Fourth dimension

Overview
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, four-dimensional space ("4D") is an abstract concept derived by generalizing the rules of three-dimensional space
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

. It has been studied by mathematicians and philosophers for almost three hundred years, both for its own interest and for the insights it offered into mathematics and related fields.

Algebraically it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions.
Discussion

Recent Discussions
Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, four-dimensional space ("4D") is an abstract concept derived by generalizing the rules of three-dimensional space
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

. It has been studied by mathematicians and philosophers for almost three hundred years, both for its own interest and for the insights it offered into mathematics and related fields.

Algebraically it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions. In particular a vector with four elements (a 4-tuple
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...

) can be used to represent a position in four-dimensional space. The space is a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, so has a metric
Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

and norm, and so all directions are treated as the same: the additional dimension is indistinguishable from the other three.

In modern physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, space and time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....

are unified in a four-dimensional Minkowski
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

continuum called spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

, whose metric
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

treats the time dimension differently from the three spatial dimensions. Spacetime is thus not a Euclidean space.

## History

The possibility of spaces with dimensions higher than three was first studied by mathematicians in the 19th century. In 1827 Möbius
August Ferdinand Möbius
August Ferdinand Möbius was a German mathematician and theoretical astronomer.He is best known for his discovery of the Möbius strip, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space. It was independently discovered by Johann Benedict...

realized that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image, and by 1853 Ludwig Schläfli
Ludwig Schläfli
Ludwig Schläfli was a Swiss geometer and complex analyst who was one of the key figures in developing the notion of higher dimensional spaces. The concept of multidimensionality has since come to play a pivotal role in physics, and is a common element in science fiction...

Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...

s in higher dimensions, although his work was not published until after his death. Higher dimensions were soon put on firm footing by Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

's 1854 Habilitationsschrift, Über die Hypothesen welche der Geometrie zu Grunde liegen, in which he considered a "point" to be any sequence of coordinates (x1, ..., xn). The possibility of geometry in higher dimensions, including four dimensions in particular, was thus established.

An arithmetic of four dimensions called quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

s was defined by William Rowan Hamilton
William Rowan Hamilton
Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...

in 1843. This associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...

was the source of the science of vector analysis in three dimensions as recounted in A History of Vector Analysis
A History of Vector Analysis
A History of Vector Analysis is a book on the history of vector analysis by Michael J. Crowe, originally published by the University of Notre Dame Press....

.

The fourth dimension was popularised by Charles Howard Hinton
Charles Howard Hinton
Charles Howard Hinton was a British mathematician and writer of science fiction works titled Scientific Romances. He was interested in higher dimensions, particularly the fourth dimension, and is known for coining the word tesseract and for his work on methods of visualising the geometry of...

, starting in 1880 with his essay What is the Fourth Dimension? published in the Dublin University magazine. He coined the terms tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...

, ana and kata in his book A New Era of Thought
A New Era of Thought
A New Era of Thought is a non-fiction work written by Charles Howard Hinton, was published in 1888 and reprinted in 1900 by Swan Sonnenschein & Co. Ltd., London. A New Era of Thought is about the fourth dimension and its implications on human thinking. It influenced the work of P.D. Ouspensky,...

, and introduced a method for visualising the fourth dimension using cubes in the book Fourth Dimension.

In 1908, Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...

presented a paper consolidating the role of time as the fourth dimension of spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

, the basis for Einstein's
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

theories of special
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

and general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

. But the geometry of spacetime, being non-Euclidean, is completely different to that popularised by Hinton. The study of such Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

s required new mathematics quite different to that of four-dimensional Euclidean space, and so developed along quite different lines. This separation was less clear in the popular imagination, with works of fiction and philosophy blurring the distinction, so in 1973 H. S. M. Coxeter felt compelled to write:

## Vectors

Mathematically four-dimensional space is simply a space with four spatial dimensions, that is a space that needs four parameters to specify a point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

in it. For example a general point might have position vector a, equal to

This can be written in terms of the four standard basis
Standard basis
In mathematics, the standard basis for a Euclidean space consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system...

vectors (e1, e2, e3, e4), given by
so the general vector a is

Vectors add, subtract and scale as in three dimensions.

The dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

of Euclidean three-dimensional space generalizes to four dimensions as

It can be used to calculate the norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

or length
Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

of a vector,
and calculate or define the angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

between two vectors as

Minkowski spacetime is four-dimensional space with geometry defined by a nondegenerate pairing
Pairing
The concept of pairing treated here occurs in mathematics.-Definition:Let R be a commutative ring with unity, and let M, N and L be three R-modules.A pairing is any R-bilinear map e:M \times N \to L...

different from the dot product:

As an example, the distance squared between the points (0,0,0,0) and (1,1,1,0) is 3 in both the Euclidean and Minkowskian 4-spaces, while the distance squared between (0,0,0,0) and (1,1,1,1) is 4 in Euclidean space and 2 in Minkowski space; increasing actually decreases the metric distance. This leads to many of the well known apparent "paradoxes" of relativity.

The cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

is not defined in four dimensions. Instead the exterior product is used for some applications, and is defined as follows:

This is bivector
Bivector
In mathematics, a bivector or 2-vector is a quantity in geometric algebra or exterior algebra that generalises the idea of a vector. If a scalar is considered a zero dimensional quantity, and a vector is a one dimensional quantity, then a bivector can be thought of as two dimensional. Bivectors...

valued, with bivectors in four dimensions forming a six-dimensional linear space with basis (e12, e13, e14, e23, e24, e34). They can be used to generate rotations in four dimensions.

## Orthogonality and vocabulary

In the familiar 3-dimensional space that we live in there are three coordinate axes — usually labeled x, y, and z — with each axis orthogonal (i.e. perpendicular) to the other two. The six cardinal directions in this space can be called up, down, east, west, north, and south. Positions along these axes can be called altitude, longitude, and latitude. Lengths measured along these axes can be called height, width, and depth.

Comparatively, 4-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled w. To describe the two additional cardinal directions, Charles Howard Hinton
Charles Howard Hinton
Charles Howard Hinton was a British mathematician and writer of science fiction works titled Scientific Romances. He was interested in higher dimensions, particularly the fourth dimension, and is known for coining the word tesseract and for his work on methods of visualising the geometry of...

coined the terms ana and kata, from the Greek words meaning "up toward" and "down from", respectively. A length measured along the w axis can be called spissitude, as coined by Henry More
Henry More
Henry More FRS was an English philosopher of the Cambridge Platonist school.-Biography:Henry was born at Grantham and was schooled at The King's School, Grantham and at Eton College...

.

## Geometry

The geometry of 4-dimensional space is much more complex than that of 3-dimensional space, due to the extra degree of freedom.

Just as in 3 dimensions there are polyhedra
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

s, in 4 dimensions there are polychora
Polychoron
In geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces , and cells...

(4-polytope
Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...

s) made of polyhedra. In 3 dimensions there are 5 regular polyhedra known as the Platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

s. In 4 dimensions there are 6 convex regular polychora, the analogues of the Platonic solids. Relaxing the conditions for regularity generates a further 58 convex uniform polychora
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....

, analogous to the 13 semi-regular Archimedean solid
Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...

s in three dimensions.

In 3 dimensions, a circle may be extruded to form a cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

. In 4 dimensions, there are several different cylinder-like objects. A sphere may be extruded to obtain a spherical cylinder (a cylinder with spherical "caps"), and a cylinder may be extruded to obtain a cylindrical prism. The Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

of two circles may be taken to obtain a duocylinder. All three can "roll" in 4-dimensional space, each with its own properties.

In 3 dimensions, curves can form knot
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...

s but surfaces cannot (unless they are self-intersecting). In 4 dimensions, however, knots made using curves can be trivially untied by displacing them in the fourth direction, but 2-dimensional surfaces can form non-trivial, non-self-intersecting knots in 4-dimensional space. Because these surfaces are 2-dimensional, they can form much more complex knots than strings in 3-dimensional space can. The Klein bottle
Klein bottle
In mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...

is an example of such a knotted surface. Another such surface is the real projective plane
Real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...

.

### Hypersphere

The set of points in Euclidean 4-space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

having the same distance R from a fixed point P0 forms a hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...

known as a 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...

. The hyper-volume of the enclosed space is:

This is part of the Friedmann–Lemaître–Robertson–Walker metric in General relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

where R is substituted by function R(t) with t meaning the cosmological age of the universe. Growing or shrinking R with time means expanding or collapsing universe, depending on the mass density inside.

## Cognition

Research using virtual reality
Virtual reality
Virtual reality , also known as virtuality, is a term that applies to computer-simulated environments that can simulate physical presence in places in the real world, as well as in imaginary worlds...

finds that humans in spite of living in a three-dimensional world can without special practice make spatial judgments based on the length of, and angle between, line segments embedded in four-dimensional space. The researchers noted that "the participants in our study had minimal practice in these tasks, and it remains an open question whether it is possible to obtain more sustainable, definitive, and richer 4D representations with increased perceptual experience in 4D virtual environments." In another study, the ability of humans to orient themselves in 2D, 3D and 4D mazes has been tested. Each maze consisted of four path segments of random length and connected with orthogonal random bends, but without branches or loops (i.e. actually labyrinth
Labyrinth
In Greek mythology, the Labyrinth was an elaborate structure designed and built by the legendary artificer Daedalus for King Minos of Crete at Knossos...

s). The graphical interface was based on John McIntosh's free 4D Maze game. The participating persons had to navigate through the path and finally estimate the linear direction back to the starting point. The researchers found that some of the participants were able to mentally integrate their path after some practice in 4D (the lower dimensional cases were for comparison and for the participants to learn the method).

## Dimensional analogy

To understand the nature of four-dimensional space, a device called dimensional analogy is commonly employed. Dimensional analogy is the study of how (n − 1) dimensions relate to n dimensions, and then inferring how n dimensions would relate to (n + 1) dimensions.

Dimensional analogy was used by Edwin Abbott Abbott
Edwin Abbott Abbott
Edwin Abbott Abbott , English schoolmaster and theologian, is best known as the author of the satirical novella Flatland .-Biography:...

in the book Flatland
Flatland
Flatland: A Romance of Many Dimensions is an 1884 satirical novella by the English schoolmaster Edwin Abbott Abbott. Writing pseudonymously as "A Square", Abbott used the fictional two-dimensional world of Flatland to offer pointed observations on the social hierarchy of Victorian culture...

, which narrates a story about a square that lives in a two-dimensional world, like the surface of a piece of paper. From the perspective of this square, a three-dimensional being has seemingly god-like powers, such as ability to remove objects from a safe without breaking it open (by moving them across the third dimension), to see everything that from the two-dimensional perspective is enclosed behind walls, and to remain completely invisible by standing a few inches away in the third dimension.

By applying dimensional analogy, one can infer that a four-dimensional being would be capable of similar feats from our three-dimensional perspective. Rudy Rucker
Rudy Rucker
Rudolf von Bitter Rucker is an American mathematician, computer scientist, science fiction author, and philosopher, and is one of the founders of the cyberpunk literary movement. The author of both fiction and non-fiction, he is best known for the novels in the Ware Tetralogy, the first two of...

illustrates this in his novel Spaceland, in which the protagonist encounters four-dimensional beings who demonstrate such powers.

### Projections

A useful application of dimensional analogy in visualizing the fourth dimension is in projection
Graphical projection
Graphical projection is a protocol by which an image of a three-dimensional object is projected onto a planar surface without the aid of mathematical calculation, used in technical drawing.- Overview :...

. A projection is a way for representing an n-dimensional object in n − 1 dimensions. For instance, computer screens are two-dimensional, and all the photographs of three-dimensional people, places and things are represented in two dimensions by projecting the objects onto a flat surface. When this is done, depth is removed and replaced with indirect information. The retina
Retina
The vertebrate retina is a light-sensitive tissue lining the inner surface of the eye. The optics of the eye create an image of the visual world on the retina, which serves much the same function as the film in a camera. Light striking the retina initiates a cascade of chemical and electrical...

of the eye
Human eye
The human eye is an organ which reacts to light for several purposes. As a conscious sense organ, the eye allows vision. Rod and cone cells in the retina allow conscious light perception and vision including color differentiation and the perception of depth...

is also a two-dimensional array of receptor
Sensory receptor
In a sensory system, a sensory receptor is a sensory nerve ending that responds to a stimulus in the internal or external environment of an organism...

s but the brain is able to perceive the nature of three-dimensional objects by inference from indirect information (such as shading, foreshortening, binocular vision
Binocular vision
Binocular vision is vision in which both eyes are used together. The word binocular comes from two Latin roots, bini for double, and oculus for eye. Having two eyes confers at least four advantages over having one. First, it gives a creature a spare eye in case one is damaged. Second, it gives a...

, etc.). Artist
Artist
An artist is a person engaged in one or more of any of a broad spectrum of activities related to creating art, practicing the arts and/or demonstrating an art. The common usage in both everyday speech and academic discourse is a practitioner in the visual arts only...

s often use perspective
Perspective (graphical)
Perspective in the graphic arts, such as drawing, is an approximate representation, on a flat surface , of an image as it is seen by the eye...

to give an illusion of three-dimensional depth to two-dimensional pictures.

Similarly, objects in the fourth dimension can be mathematically projected to the familiar 3 dimensions, where they can be more conveniently examined. In this case, the 'retina' of the four-dimensional eye is a three-dimensional array of receptors. A hypothetical being with such an eye would perceive the nature of four-dimensional objects by inferring four-dimensional depth from indirect information in the three-dimensional images in its retina.

The perspective projection of three-dimensional objects into the retina of the eye introduces artifacts such as foreshortening, which the brain interprets as depth in the third dimension. In the same way, perspective projection from four dimensions produces similar foreshortening effects. By applying dimensional analogy, one may infer four-dimensional "depth" from these effects.

As an illustration of this principle, the following sequence of images compares various views of the 3-dimensional cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

with analogous projections of the 4-dimensional tesseract into three-dimensional space.
Cube Tesseract Description
The image on the left is a cube viewed face-on. The analogous viewpoint of the tesseract in 4 dimensions is the cell-first perspective projection, shown on the right. One may draw an analogy between the two: just as the cube projects to a square, the tesseract projects to a cube.

Note that the other 5 faces of the cube are not seen here. They are obscured by the visible face. Similarly, the other 7 cells of the tesseract are not seen here because they are obscured by the visible cell.
The image on the left shows the same cube viewed edge-on. The analogous viewpoint of a tesseract is the face-first perspective projection, shown on the right. Just as the edge-first projection of the cube consists of two trapezoid
Trapezoid
In Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...

s, the face-first projection of the tesseract consists of two frustum
Frustum
In geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....

s.
The nearest edge of the cube in this viewpoint is the one lying between the red and green faces. Likewise, the nearest face of the tesseract is the one lying between the red and green cells.
On the left is the cube viewed corner-first. This is analogous to the edge-first perspective projection of the tesseract, shown on the right. Just as the cube's vertex-first projection consists of 3 deltoids
Kite (geometry)
In Euclidean geometry a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are next to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite each other rather than next to each other...

surrounding a vertex, the tesseract's edge-first projection consists of 3 hexahedral
Hexahedron
A hexahedron is any polyhedron with six faces, although usually implies the cube as a regular hexahedron with all its faces square, and three squares around each vertex....

volumes surrounding an edge. Just as the nearest vertex of the cube is the one where the three faces meet, so the nearest edge of the tesseract is the one in the center of the projection volume, where the three cells meet.
A different analogy may be drawn between the edge-first projection of the tesseract and the edge-first projection of the cube. The cube's edge-first projection has two trapezoids surrounding an edge, while the tesseract has three hexahedral volumes surrounding an edge.
On the left is the cube viewed corner-first. The vertex-first perspective projection of the tesseract is shown on the right. The cube's vertex-first projection has three tetragons surrounding a vertex, but the tesseract's vertex-first projection has four hexahedral volumes surrounding a vertex. Just as the nearest corner of the cube is the one lying at the center of the image, so the nearest vertex of the tesseract lies not on boundary of the projected volume, but at its center inside, where all four cells meet.
Note that only three faces of the cube's 6 faces can be seen here, because the other 3 lie behind these three faces, on the opposite side of the cube. Similarly, only 4 of the tesseract's 8 cells can be seen here; the remaining 4 lie behind these 4 in the fourth direction, on the far side of the tesseract.

A concept closely related to projection is the casting of shadows.

If a light is shone on a three dimensional object, a two-dimensional shadow is cast. By dimensional analogy, light shone on a two-dimensional object in a two-dimensional world would cast a one-dimensional shadow, and light on a one-dimensional object in a one-dimensional world would cast a zero-dimensional shadow, that is, a point of non-light. Going the other way, one may infer that light shone on a four-dimensional object in a four-dimensional world would cast a three-dimensional shadow.

If the wireframe of a cube is lit from above, the resulting shadow is a square within a square with the corresponding corners connected. Similarly, if the wireframe of a tesseract were lit from “above” (in the fourth direction), its shadow would be that of a three-dimensional cube within another three-dimensional cube. (Note that, technically, the visual representation shown here is actually a two-dimensional shadow of the three-dimensional shadow of the four-dimensional wireframe figure.)

### Bounding volumes

Dimensional analogy also helps in inferring basic properties of objects in higher dimensions. For example, two-dimensional objects are bounded by one-dimensional boundaries: a square is bounded by four edges. Three-dimensional objects are bounded by two-dimensional surfaces: a cube is bounded by 6 square faces. By applying dimensional analogy, one may infer that a four-dimensional cube, known as a tesseract
Tesseract
In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...

, is bounded by three-dimensional volumes. And indeed, this is the case: mathematics shows that the tesseract is bounded by 8 cubes. Knowing this is key to understanding how to interpret a three-dimensional projection of the tesseract. The boundaries of the tesseract project to volumes in the image, not merely two-dimensional surfaces.

### Visual scope

Being three-dimensional, we are only able to see the world with our eyes in two dimensions. A four-dimensional being would be able to see the world in three dimensions. For example, it would be able to see all six sides of an opaque box simultaneously, and in fact, what is inside the box at the same time, just as we can see the interior of a square on a piece of paper. It would be able to see all points in 3-dimensional space simultaneously, including the inner structure of solid objects and things obscured from our three-dimensional viewpoint.

### Limitations

Reasoning by analogy from familiar lower dimensions can be an excellent intuitive guide, but care must be exercised not to accept results that are not more rigorously tested. For example, consider the formulas for the circumference of a circle

and the surface area of a sphere:
.
One might be tempted to suppose that the surface volume of a hypersphere is , or perhaps , but either of these would be wrong. The correct formula is .